Theorem subexinf-P5 608

Description: Inference Version of '∃𝑥' Substitution Law.

For the deductive form with a variable restriction see subexv-P5 624. For the most general form see subex-P6 754.

Hypothesis

Ref	Expression
subexinf-P5.1	⊢ (𝜑 ↔ 𝜓)

Assertion

Ref	Expression
subexinf-P5	⊢ (∃𝑥𝜑 ↔ ∃𝑥𝜓)

Proof of Theorem subexinf-P5

Step	Hyp	Ref	Expression
1		subexinf-P5.1	. . . 4 ⊢ (𝜑 ↔ 𝜓)
2	1	rcp-NDBIEF0 240	. . 3 ⊢ (𝜑 → 𝜓)
3	2	alloverimex-P5.RC.GEN 603	. 2 ⊢ (∃𝑥𝜑 → ∃𝑥𝜓)
4	1	rcp-NDBIER0 241	. . 3 ⊢ (𝜓 → 𝜑)
5	4	alloverimex-P5.RC.GEN 603	. 2 ⊢ (∃𝑥𝜓 → ∃𝑥𝜑)
6	3, 5	rcp-NDBII0 239	1 ⊢ (∃𝑥𝜑 ↔ ∃𝑥𝜓)

Syntax hints:   ↔ 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

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-E5.04a  675  exi-P6  718  qcexandr-P6  761  qcexandl-P6  762  psubneg-P6-L1  787  psubspliteq-P6  800  psubsplitelof-P6  801