Theorem subexv-P5 624

Description: Substitution Law for '∃𝑥' (variable restriction). The most general form is subex-P6 754.

Hypothesis

Ref	Expression
subexv-P5.1	⊢ (𝛾 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
subexv-P5	⊢ (𝛾 → (∃𝑥𝜑 ↔ ∃𝑥𝜓))

Distinct variable group:   𝛾,𝑥

Proof of Theorem subexv-P5

Step	Hyp	Ref	Expression
1		subexv-P5.1	. . . 4 ⊢ (𝛾 → (𝜑 ↔ 𝜓))
2	1	ndbief-P3.14 179	. . 3 ⊢ (𝛾 → (𝜑 → 𝜓))
3	2	alloverimex-P5.GENV 622	. 2 ⊢ (𝛾 → (∃𝑥𝜑 → ∃𝑥𝜓))
4	1	ndbier-P3.15 180	. . 3 ⊢ (𝛾 → (𝜓 → 𝜑))
5	4	alloverimex-P5.GENV 622	. 2 ⊢ (𝛾 → (∃𝑥𝜓 → ∃𝑥𝜑))
6	3, 5	ndbii-P3.13 178	1 ⊢ (𝛾 → (∃𝑥𝜑 ↔ ∃𝑥𝜓))

Syntax hints:   → wff-imp 10   ↔ wff-bi 104  ∃wff-exists 595

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

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

This theorem is referenced by:  example-E5.02a  664  example-E5.04a  675  nfrex2w-P6  695  example-E6.02a  712