Theorem bnj1198 32062

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj1198.1	⊢ (𝜑 → ∃𝑥𝜓)
bnj1198.2	⊢ (𝜓′ ↔ 𝜓)

Assertion

Ref	Expression
bnj1198	⊢ (𝜑 → ∃𝑥𝜓′)

Proof of Theorem bnj1198

Step	Hyp	Ref	Expression
1		bnj1198.1	. 2 ⊢ (𝜑 → ∃𝑥𝜓)
2		bnj1198.2	. . 3 ⊢ (𝜓′ ↔ 𝜓)
3	2	exbii 1844	. 2 ⊢ (∃𝑥𝜓′ ↔ ∃𝑥𝜓)
4	1, 3	sylibr 236	1 ⊢ (𝜑 → ∃𝑥𝜓′)

Syntax hints:   → wi 4   ↔ wb 208  ∃wex 1776

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806

This theorem depends on definitions:  df-bi 209  df-ex 1777

This theorem is referenced by:  bnj1209  32063  bnj1275  32080  bnj1340  32090  bnj1345  32091  bnj605  32174  bnj607  32183  bnj906  32197  bnj908  32198  bnj1189  32276  bnj1450  32317  bnj1312  32325