Theorem ndnfrleq-P7.11 836

Description: Natural Deduction: Effective Non-Freeness Rule 11.

If '𝑥' is not effectively free in the context '𝛾', then the (conditional) effective non-freeness of '𝑥' is conserved between (conditionally) logically equivalent WFFs.

Hypotheses

Ref	Expression
ndnfrleq-P7.11.1	⊢ Ⅎ𝑥𝛾
ndnfrleq-P7.11.2	⊢ (𝛾 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
ndnfrleq-P7.11	⊢ (𝛾 → (Ⅎ𝑥𝜑 ↔ Ⅎ𝑥𝜓))

Proof of Theorem ndnfrleq-P7.11

Step	Hyp	Ref	Expression
1		ndnfrleq-P7.11.1	. 2 ⊢ Ⅎ𝑥𝛾
2		ndnfrleq-P7.11.2	. 2 ⊢ (𝛾 → (𝜑 ↔ 𝜓))
3	1, 2	subnfr-P6 755	1 ⊢ (𝛾 → (Ⅎ𝑥𝜑 ↔ Ⅎ𝑥𝜓))

Syntax hints:   → wff-imp 10   ↔ wff-bi 104  Ⅎwff-nfree 681

This theorem was proved from axioms:  ax-L1 11  ax-L2 12  ax-L3 13  ax-MP 14  ax-GEN 15  ax-L4 16  ax-L5 17  ax-L6 18  ax-L7 19  ax-L12 29

This theorem depends on definitions:  df-bi-D2.1 107  df-and-D2.2 133  df-or-D2.3 145  df-true-D2.4 155  df-rcp-AND3 161  df-exists-D5.1 596  df-nfree-D6.1 682

This theorem is referenced by:  ndnfrleq-P7.11.VR  862  gennfrcl-P7  963  nfrsucc-P8  1119  nfradd-P8  1120  nfrmult-P8  1121